1. As mentioned in the comments of Itô's Formula for functions that are C2 almost everywhere

in "Ito's Formula for Non-Smooth Functions", they prove a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

2. In the post https://math.stackexchange.com/questions/2521208/other-versions-of-a-weak-ito-formula, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.

3. In the post Generalized Ito's formula, they further give the following two references:

• Krylov's "Controlled Diffusion Processes" Ch 2 Section 10.
• H. F"ollmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1-20 (2000). Consider a d-dimensional Brownian motion X = (X1,...,Xd ) and a function F which belongs locally to the Sobolev space $W^{1,2}$. We prove an extension of Ito’s formula ˆ where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of F.

4. Some other extensions (eg. convex F) are listed here https://math.stackexchange.com/questions/4803597/can-we-apply-ito-formula-to-quadratic-variation-of-c1-function-of-semimarting.

1. As mentioned by Mateusz Kwaśnicki in the comments, in "Itô's Formula for Non-Smooth Functions", Aebi proves a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

2. In the post Other versions of a weak Ito formula?, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.

3. In the post Generalized Ito's formula, they further give the following two references:

• Gerard referenced Krylov's "Controlled Diffusion Processes" Ch. 2 Section 10.
• Anatoly Kochubei referenced H. Föllmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1–20 (2000). Consider a $d$-dimensional Brownian motion $X = (X_1,\dotsc,X_d )$ and a function $F$ which belongs locally to the Sobolev space $W^{1,2}$. We prove an extension of Itô’s formula where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of $F$.

4. Some other extensions (eg. convex $F$) are listed here Can we apply Ito formula to quadratic variation of $C^1$ function of semimartingale?.

1. As mentioned in the comments of Itô's Formula for functions that are C2 almost everywhere

in "Ito's Formula for Non-Smooth Functions", they prove a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

2. In the post https://math.stackexchange.com/questions/2521208/other-versions-of-a-weak-ito-formula, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.

3. In the post Generalized Ito's formula, they further give the following two references:

• Krylov's "Controlled Diffusion Processes" Ch 2 Section 10.
• H. F"ollmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1-20 (2000). Consider a d-dimensional Brownian motion X = (X1,...,Xd ) and a function F which belongs locally to the Sobolev space $W^{1,2}$. We prove an extension of Ito’s formula ˆ where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of F.

1. As mentioned in the comments of Itô's Formula for functions that are C2 almost everywhere

in "Ito's Formula for Non-Smooth Functions", they prove a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

2. In the post https://math.stackexchange.com/questions/2521208/other-versions-of-a-weak-ito-formula, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.

3. In the post Generalized Ito's formula, they further give the following two references:

• Krylov's "Controlled Diffusion Processes" Ch 2 Section 10.
• H. F"ollmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1-20 (2000). Consider a d-dimensional Brownian motion X = (X1,...,Xd ) and a function F which belongs locally to the Sobolev space $W^{1,2}$. We prove an extension of Ito’s formula ˆ where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of F.

4. Some other extensions (eg. convex F) are listed here https://math.stackexchange.com/questions/4803597/can-we-apply-ito-formula-to-quadratic-variation-of-c1-function-of-semimarting.